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\(N\)-dimensional shape optimization under capacitary constraint. (English) Zbl 0847.49029
The shape optimization problem
Minimize \[ J(\Omega)= \textstyle{{1\over 2}} \int_B (u_\Omega- g)^2 dx,\tag{1} \] subject to \[ - {\mathcal A} u_\Omega= f\tag{2} \] is considered. The following notations are chosen:
Let \(B\subset \mathbb{R}^N\) be an open ball, \(A\in M_{n\times n}(C^1(\overline B))\), \(A= A^*\), \(\alpha I\leq A\leq \beta I\), \(0< \alpha< \beta\). Further, the associated operator \({\mathcal A}: H^1_0(B)\to H^{- 1}(B)\) with \({\mathcal A}= \text{div}(A\nabla)\) is defined. \(f\in H^{- 1}(B)\) and the open subset \(\Omega\) of \(B\) are considered. The state equation (2) has to be understood in the variational sense.
The existence of extremal domains of problem (1)–(2) is investigated. Especially, relations between finding of compact sets in some topology on the space of domains and the continuity of the map \(\Omega\mapsto J(\Omega)\) is discussed.
The authors obtain results for the \(N\)-dimensional case for classes of domains satisfying capacity density conditions.

49Q10 Optimization of shapes other than minimal surfaces
49J20 Existence theories for optimal control problems involving partial differential equations
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